A fence 2.4 m tall, is 10m away from a tree of height 16m. Calculate the angle of elevation of the top of the tree from the top of the fence.

Analyze the question

The fence is 2.4 m tall and stands 10 m from the tree, which is 16 m tall. We want the angle of elevation of the tree's top, seen from the top of the fence.

Concept

Angle of elevation uses the right-angled triangle formula tan θ = opposite / adjacent. The observer's eye level is the top of the fence, so only the height of the tree above the fence counts as the opposite side.

Solution

1. Height of the tree above the fence top: 16 − 2.4 = 13.6 m (opposite)
2. Horizontal distance: 10 m (adjacent)
3. tan θ = 13.6 / 10 = 1.36
4. θ = tan⁻¹(1.36)

tan 53° ≈ 1.327 and tan 54° ≈ 1.376. Interpolating, (1.36 − 1.327)/(1.376 − 1.327) ≈ 0.67, so θ ≈ 53.67°.

Answer

θ ≈ 53.67°, which is option B.

Tip

In angle of elevation questions, always subtract the observer's height from the object's height before using tan. Many students wrongly use 16/10, which gives about 58°, a trap answer. Also, 76.11° comes from tan⁻¹(4) and is not related to this question.